TWENTYSECOND GÖKOVA GEOMETRY / TOPOLOGY CONFERENCE
May 25  May 30 (2015)
Gökova, Turkey
List of invited speakers/participants
C. Manolescu  
E. Murphy  
D. Ruberman 
F. Lin  
M. Kreck  
D. Auckly 
A. Wand  
A. Stipsicz  
G. Dimitroglou Rizell 
P. Ghiggini  
J. Nelson  
B. Chantraine 
I. Petkova  
S. Galkin  
C. Blanchet 
B. Ozbagci  
T. Dereli  
S. Finashin 
M. Bhupal  
M. Kalafat  
I. Unal 
 
S. Uguz  

Scientific Committee : D. Auroux, Y. Eliashberg, I. Itenberg, G. Mikhalkin, S. Akbulut
Organizing Commitee : T. Önder, S. Koçak, A. Degtyarev, Ö. Kişisel, Y. Ozan, T. Etgü, S. Salur
Supporting Organizations: NSF (National Science Foundation)
and
TMD (Turkish Mathematical Society).
The participants of 22^{nd} Gökova Geometry  Topology Conference
List of Talks

Ciprian Manolescu  
Mini course: The triangulation conjecture and related results
The triangulation conjecture stated that any ndimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. The goal of this minicourse is to explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of GalewskiStern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3dimensional homology cobordism group). The lowdimensional question can be answered in the negative using a variant of Floer homology, Pin(2)equivariant SeibergWitten Floer homology. Time permitting, I will also discuss other related developments: an application of Pin(2)equivariant SeibergWitten Floer Ktheory (to the intersection forms of spin fourmanifolds with boundary), and a construction (joint with Kristen Hendricks) of an involutive variant of Heegaard Floer homology  the analog of Z_{4}equivariant SeibergWitten Floer homology.


Emmy Murphy  
Mini course: Geometry of contact structures and overtwistedness in high
dimensions
We discuss a number of geometric pictures of contact
manifolds, focusing on interactions of these pictures with the recent
concept of overtwistedness. The main topics we will cover are:
Legendrian manifolds and looseness, Weinstein cobordisms and contact
surgery, the plastikstufe and fillability, and open book
decompositions. Many of these ideas have been around for a long time.
However the relationships between these pictures have only begun to
take shape, spurred on by the recent discovery of overtwistedness.


Daniel Ruberman  
Absolutely exotic 4manifolds
We show the existence of exotic smooth structures on contractible 4manifolds. These structures are absolute, in the sense that they do not depend on a specific marking of the boundary. This is in contrast to the phenomenon of corks, which are exotic relative to an automorphism of their boundaries. The technique is to modify a relatively exotic manifold to give an exotic one for which we have a good understanding of the automorphism group of the boundary. This is joint work with Selman Akbulut.


Jo Nelson  
An integral lift of contact homology
Cylindrical contact homology is arguably one of the more notorious Floertheoretic constructions. The past decade has been less than kind to this theory, as the growing knowledge of gaps in its foundations has tarnished its claim to being a welldefined contact invariant. However, recent work of Hutchings and Nelson has managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. This talk will highlight our implementation of nonequivariant constructions, domain dependent almost complex structures, automatic transversality, and obstruction bundle gluing, yielding a homological contact invariant which is expected to be isomorphic to SH^{+} under suitable assumptions, though it does not require a filling of the contact manifold. By making use of family Floer theory we obtain an S^{1}equivariant theory defined over Z coefficients, which when tensored with Q yields cylindrical contact homology, now with the guarantee of welldefinedness and invariance.


Ina Petkova  
Combinatorial tangle Floer homology
In joint work with Vera Vertesi, we extend the functoriality in Heegaard Floer homology by defining a Heegaard Floer invariant for tangles which satisfies a nice gluing formula. We will discuss the construction of this combinatorial invariant for tangles in S^{3}, D^{3}, and I x S^{2}. The special case of S^{3} gives back a stabilized version of knot Floer homology. If time permits, I will discuss how tangle Floer homology enhances the structure of knot Floer homology, focusing on its gluing properties, its similarities with other theories, and its relation to quantum topology.


Matthias Kreck  
Manifolds which are like the associative Grassmannian (joint work with D. Crowley and D. Salamon)
The associative Grassmannian is the homogeneous space G_{2}/SO(4), where G_{2} is the exceptional Lie group of dimension 14. Akbulut and Kalafat have determined the cohomology ring which gives a rational homology quaternionic projective plane. Eells and Kuiper have written a paper "Manifolds which are like the projective plane" and investigated the diffeomorphism type of simply connected manifolds which have the cohomology ring of the projective planes. We do the same with the associative Grassmannian instead of the projective plane.


Andras Stipsicz  
Knot Floer homologies
Knot Floer homology (introduced by OzsvathSzabo and independently by
Rasmussen) is a powerful tool for studying knots and links in the 3sphere. In
particular, it gives rise to a numerical invariant, which provides a
nontrivial lower bound on the 4dimensional genus of the knot. By deforming
the definition of knot Floer homology by a real number t from [0,2], we define
a family of homologies, and derive a family of numerical invariants with
similar properties. The resulting invariants provide a family of
homomorphisms on the concordance group. One of these homomorphisms can be
used to estimate the unoriented 4dimensional genus of the knot. We will
review the basic constructions for knot Floer homology and the deformed
theories and discuss some of the applications. This is joint work with
P. Ozsvath and Z. Szabo.


Georgios Dimitroglou Rizell  
The classification of Lagrangian tori in a fourdimensional symplectic vectorspace
We prove that there are exactly two monotone Lagrangian tori in a fourdimensional symplectic vectorspace up to Hamiltonian isotopy and rescaling: the Clifford torus and the Chekanov torus. Moreover, all nonmonotone tori are shown to be Hamiltonian isotopic to product tori. The strategy is, first, finding a singular symplectic conic linking the torus appropriately and, second, applying a classification result for homotopically nontrivial Lagrangian tori inside the cotangent bundle of a twotorus. The latter result follows using methods due to Ivrii.


Andy Wand  
Tightness and open book decompositions
We will discuss a characterization of tightness of a contact 3manifold in terms of supporting open book decompositions, some applications, and generalizations.


Francesco Lin  
The surgery exact triangle in Pin(2)monopole Floer homology
In this talk we discuss an approach to Manolescu's Pin(2)equivariant SeibergWitten Floer homology
from Kronheimer and Mrowka's Morse theoretic point of view, and discuss the relationship between
the invariants associated to surgeries on a given knot.


Paolo Ghiggini  
Tight contact structures on connected sums which are not contact sonnected sum
It is well known that, in dimension three, every tight contact structure on a connected sum
is a contact connected sum. I will show that the same statement is not true in dimension five.
This is a joint work with Klaus Niederkrüger and Chris Wendl.


Christian Blanchet  
Non semisimple TQFTs from nilpotent representations of quantum sl(2)
A new family of quantum invariants based on nilpotent representions of quantum sl(2)
at a root of unity have been constructed by CostantinoGeerPatureau. We show that the
new quantum invariants have graded TQFT extensions. We will consider the specific case of
root of unity of order 4 where CGP invariants recover Reidemeister torsion with canonical
normalisation. In the general case we will describe the arising representations of mapping class groups.
(Joint work with François Costantino, Nathan Geer and Bertrand Patureau) 

